Apple TV gives up on WiFi association after some time

You say you "disconnect the WiFi", but I think you mean that you turn it off at the router. ATV continues trying to connect, but gives up after a while. Having given up, it won't try again unless you interact with it somehow. I suspect that it's designed to do that. It's not surprising that it remembers the network key, since I think it behaves like an iPad or iPhone, saving the SSIDs and keys of networks it has previously seen.

You can try complaining to Apple at http://www.apple.com/feedback/appletv.html but I suspect this requirement was not included in the design. After all, ATV was really designed for home use, as indicated by the "watch TV" apps. Schools and businesses find them useful, but that's not the primary product focus.

Why is it necessary to turn the WiFi off?

Could you use a wired connection for ATV? Assuming your router supports it, wireless clients could still AirPlay to ATV.

I was going to suggest that you enable conference room mode on ATV, but it sounds like you're already doing that.

It's a little brutal, but could you power down ATV (pull the plug) before turning off WiFi? That way, ATV would never know that WiFi goes away.

By the way, there are other reports of ATV apparently giving up on network connections while sleeping, even when WiFi remains available. These users have to do the same as you to get ATV to pay attention to networks again. The funny thing is that it doesn't affect everyone. There may be some unrelated ATV setting that triggers this behavior. So you might hope for a software fix, or one of these other threads might finally come up with a solution. Just search here for WiFi and sleep.

Did this use to work? Did it begin to give trouble after the stealth 7.2.1 refresh (same version number) on Feb 25, 2016?

This may be an intentional security feature, to prevent a hacker spoofing the prior network and invading the ATV. If it's intentional, it'll be much harder to get a fix.

I fear the other installations may just be pending updates. I guess you'd be safe if auto update is turned off, and of course you don't give your units much chance to see the internet. I asked about the update because there have been other posts about network trouble following the update.

I think you're safe with automatic updates disabled.

Since you're a large user, you may be able to get some information from Apple tech support; out of my league, I'm afraid. Certainly, there are other reports here of ATV dropping WiFi after sleep, even when the network is always available.

You have ATV 3, which is supposed to be stable now, so it's hard to say if this is a bug or a design feature. Long ago we had a DEC computer whose software didn't work as documented.. They fixed it in the next update, by sending a revised documentation page

I'm afraid not. Apple has stopped "signing" the earlier releases, so they cannot be installed. You could try an Apple store, but they probably cannot do it either. Also, just for info, note that there are two versions of the ATV hardware, see Identify your Apple TV model - Apple Support